MATHEMATICAL SYMBOLS
=
Equal sign: equality
≠
Not equal sign: inequality
≈
Approximately equal: approximation
>
Strict inequality: greater than
<
Strict inequality: less than
≥
Inequality: greater than or equal to
≤
Inequality: less than or equal to
<<
Inequality: much less than
>>
Inequality: much greater than
( )
Parentheses: calculate expression inside first
[ ]
Brackets: calculate expression inside first
+
Plus sign: addition
−
Minus sign: subtraction
±
Plus / minus: both plus and minus operations
*
Asterisk: multiplication
×
Times sign: multiplication
⋅
Multiplication dot: multiplication
÷
Division sign / obelus: division
/
Division slash: division
—
Horizontal line: division / fraction
mod
Modulo: remainder calculation
.
Period: decimal point / decimal separator
a^{b}
Power: exponent
a^b
Caret: exponent
√a
Square root: √a ⋅ √a = a
^{3}√a
Cube root: ^{3}√a ⋅ ^{3}√a ⋅ ^{3}√a = a
^{4}√a
Fourth root: ^{4}√a ⋅ ^{4}√a ⋅ ^{4}√a ⋅ ^{4}√a = a
^{n}√a
Nth root (radical)
%
Percent: 1% = 1/100
‰
Permille: 1‰ = 1/1000 = 0.1%
ppm
Permillion: 1ppm = 1/1000000
ppb
Perbillion: 1ppb = 1/1000000000
ppt
Pertrillion: 1ppt = 10^12
GEOMETRY SYMBOLS
∠
Angle: formed by two rays
∟
Right angle: 90°
°
Degree: 1 turn = 360°
deg
Degree: 1 turn = 360deg
′
Prime: arcminute, 1° = 60′
″
Double prime: arcsecond, 1′ = 60″
AB
Line segment: line from point A to point B
⊥
Perpendicular: perpendicular lines (90° angle)
∥
Parallel: parallel lines
≅
Congruent to: equivalence of geometric shapes and size
~
Similarity: same shapes, not same size
Δ
Triangle: triangle shape
xy
Distance: distance between points x and y
π
Pi constant: π = 3.141592654... is the ratio between the circumference and diameter of a circle
rad
Radians: radians angle unit
c
Radians: radians angle unit
grad
Gradians / gons: grads angle unit
g
Gradians / gons: grads angle unit
ALGEBRA SYMBOLS
x
Variable x: unknown value to find
≡
Equivalence: identical to
≜
Equal by definition
:=
Equal by definition
~
Approximately equal: weak approximation
∝
Proportional to
∞
Lemniscate: infinity symbol
≪
Much less than
≫
Much greater than
{ }
Braces: set
⌊x⌋
Floor brackets: rounds number to lower integer
⌈x⌉
Ceiling brackets: rounds number to upper integer
x!
Exclamation mark: factorial
x
Vertical bars: absolute value or modulus
f(x)
Function of x: maps values of x to f(x)
(f∘g)
Function composition: (f∘g) (x) = f(g(x))
(a,b)
Open interval: (a,b) = {x  a < x < b}
[a,b]
Closed interval: [a,b] = {x  a ≤ x ≤ b}
∆
Delta: change / difference
∆
Discriminant: Δ = b^{2}4ac
Σ
Sigma: summation / sum of all values in range of series
ΣΣ
Double sigma: double summation
∏
Capital pi: product / product of all values in range of series
e
Constant e / Euler's number: e = 2.718281828...
γ
EulerMascheroni constant: γ = 0.5772156649...
φ
Golden ratio constant
LINEAR ALGEBRA SYMBOLS
·
Dot: scalar product
×
Cross: vector product
A⊗B
Tensor product: tensor product of A and B
[ ]
Brackets: matrix of numbers
( )
Parentheses: matrix of numbers
 A 
Determinant: determinant of matrix A
det(A)
Determinant: determinant of matrix A
 x 
Double vertical bars: norm
A^{T}
Transpose: matrix transpose
A†
Hermitian matrix: matrix conjugate transpose
A*
Hermitian matrix: matrix conjugate transpose
A^{1}
Inverse matrix: A A^{1} = I
rank(A)
Matrix rank: rank of matrix A
dim(U)
Dimension: dimension of matrix A
STATISTICS SYMBOLS
P(A)
Probability function: probability of event A
P(A ⋂ B)
Probability of events intersection: probability of events A and B
P(A ⋃ B)
Probability of events union: probability of events A or B
P(A  B)
Conditional probability function: probability of event A given event B occured
f(x)
Probability density function (pdf): P(a ≤ x ≤ b) = ∫ f(x) dx
F(x)
Cumulative distribution function (cdf): F(x) = P(X ≤ x)
μ
Population mean: mean of population values
E(X)
Expectation value: expected value of random variable X
E(X  Y)
Conditional expectation: expected value of random variable X given Y
var(X)
Variance: variance of random variable X
σ^{2}
Variance: variance of population values
std(X)
Standard deviation: standard deviation of random variable X
σ_{X}
Standard deviation: standard deviation of random variable X
cov(X,Y)
Covariance: covariance of random variables X and Y
corr(X,Y)
Correlation: correlation of random variables X and Y
ρ_{X,Y}
Correlation: correlation of random variables X and Y
∑
Summation: sum of all values in range of series
∑∑
Double summation
Mo
Mode: value that occurs most frequently in population
MR
Midrange: MR = (x_{max}+x_{min})/2
Md
Sample median: half the population is below this value
Q_{1}
Lower / first quartile: 25% of population are below this value
Q_{2}
Median / second quartile: 50% of population are below this value = median of samples
Q_{3}
Upper / third quartile: 75% of population are below this value
x
Sample mean: average / arithmetic mean
s^{2}
Sample variance: population samples variance estimator
s
Sample standard deviation: population samples standard deviation estimator
z_{x}
Standard score: z_{x} = (xx) / s_{x}
X~
Distribution of X: distribution of random variable X
N(μ,σ^{2})
Normal distribution: gaussian distribution
U(a,b)
Uniform distribution: equal probability in range a,b
exp(λ)
Exponential distribution: f(x) = λe^{λx} , x ≥ 0
gamma(c,λ)
Gamma distribution: f(x) = λcx^{c1}e^{λx} / Γ(c), x ≥ 0
χ^{2}(k)
Chisquare distribution: f(x) = x^{k/21}e^{x/2} / (2^{k/2} Γ(k/2))
F(k_{1}, k_{2})
F distribution
Bin(n,p)
Binomial distribution: f(k) = _{n}C_{k} p^{k}(1p)^{nk}
Poisson(λ)
Poisson distribution: f(k) = λ^{k}e^{λ} / k!
Geom(p)
Geometric distribution: f(k) = p(1p)^{k}
HG(N,K,n)
Hypergeometric distribution
Bern(p)
Bernoulli distribution
COMBINATORICS SYMBOLS
n!
Factorial  n! = 1⋅2⋅3⋅...⋅n
_{n}P_{k}
Permutation: _{n}P_{k} = n! / (nk)!
_{n}C_{k}
Combination: _{n}C_{k} = n! / k!(nk)!
SET THEORY SYMBOLS
{ }
Set: a collection of elements
A ∩ B
Intersection: objects that belong to set A and set B
A ∪ B
Union: objects that belong to set A or set B
A ⊆ B
Subset: A is a subset of B; set A is included in set B
A ⊂ B
Proper subset / strict subset: set A is a subset of set B, and set A is not equal to set B
A ⊄ B
Not subset: set A is not a subset of set B
A ⊇ B
Superset: A is a superset of B; set A includes set B
A ⊃ B
Proper superset / strict superset: A is a superset of B, and A is not equal to B
A ⊅ B
Not superset: set A is not a superset of set B
A ⊅ B
Not superset: set A is not a superset of set B
2^{A}
Power set: all subsets of A
P(A)
Power set: all subsets of A
A = B
Equality: both sets have the same members
A^{c}
Complement: all the objects that do not belong to set A
A \ B
Relative complement: objects that belong to A and not to B
A  B
Relative complement: objects that belong to A and not to B
A ∆ B
Symmetric difference: objects that belong to A or B but not to their intersection
A ⊖ B
Symmetric difference: objects that belong to A or B but not to their intersection
a ∈ A
Element of, belongs to: set membership
x ∉ A
Not element of: no set membership
(a,b)
Ordered pair: collection of 2 elements
A×B
Cartesian product: set of all ordered pairs from A and B
A
Cardinality: the number of elements of set A
#A
Cardinality: the number of elements of set A

Vertical bar: such that
N_{0}
Alephnull: infinite cardinality of natural numbers set
N_{1}
Alephone: cardinality of countable ordinal numbers set
Ø
Empty set: Ø = { }
U
Universal set: set of all possible values
ℕ_{0}
Natural numbers with zero: ℕ_{0} = {0, 1, 2, 3, 4,...}
ℕ_{1}
Natural numbers without zero: ℕ_{1} = {1, 2, 3, 4, 5,...}
ℕ^{*}
Natural numbers without zero: ℕ^{*} = {1, 2, 3, 4, 5,...}
ℤ
Integer numbers: ℤ = {...3, 2, 1, 0, 1, 2, 3,...}
ℚ
Rational numbers: ℚ = {x  x = a/b, a,b ∈ ℤ}
ℝ
Real numbers: ℝ = {x  ∞ < x < +∞}
ℝ
Extended real numbers: ℝ = {x  ∞ ≤ x ≤ +∞}, ℝ = ℝ ∪ {∞,+∞}
ℂ
Complex numbers: ℂ = {z  z = a+bi, ∞ < a < +∞, ∞ < b < +∞}
LOGIC SYMBOLS
⋅
And
^
Caret / circumflex: and
&
Ampersand: and
+
Plus: or
∨
Reversed caret: or

Vertical line: or
x'
Single quote: not / negation
x
Bar: not / negation
¬
Not / negation
!
Exclamation mark: not / negation
⊕
Circled plus / oplus: exclusive or / xor
~
Tilde: negation
⇒
Implies
⇔
Equivalent: if and only if (iff)
↔
Equivalent: if and only if (iff)
∀
For all
∃
There exists
∃!
There exists one and only one
∄
There does not exist
∴
Therefore
∵
Because / since
∵
Because / since
CALCULUS & ANALYSIS SYMBOLS
lim_{x→a}f(x)
Limit value of a function
ε
Epsilon: represents a very small number, near zero
y'
Derivative: Lagrange's notation
y''
Second derivative: derivative of derivative
y''
Second derivative: derivative of derivative
y^{(n)}
Nth derivative: n times derivation
dy
——
dx
Derivative: Leibniz's notation
d^{2}y
——
dx^{2}
Derivative: Leibniz's notation
d^{n}y
——
dx^{n}
Nth derivative: n times derivation
D_{x}y
Derivative: Euler's notation
D_{x2}y
Second derivative: derivative of derivative
∂f(x,y)
———
∂x
Partial derivative
∫
Integral: opposite to derivation
∫∫
Double integral: integration of function of 2 variables
∫∫∫
Triple integral: integration of function of 3 variables
∮
Closed contour / line integral
∯
Closed surface integral
∰
Closed volume integral
i
Imaginary unit: i ≡ √1
z*
Complex conjugate: z = a+bi → z* = abi
z
Complex conjugate: z = a+bi → z = abi
Re(z)
Real part of a complex number: z = a+bi → Re(z) = a
Im(z)
Imaginary part of a complex number: z = a+bi → Im(z) = b
 z 
Absolute value / modulus / magnitude of a complex number: z = a+bi = √(a^{2}+b^{2})
arg(z)
Argument of a complex number: the angle of the radius in the complex plane
∇
Nabla / del: gradient / divergence operator
x * y
Convolution: y(t) = x(t) * h(t)
δ
Delta function
GREEK ALPHABET LETTERS
Α  α  Alpha
Β  β  Beta
Γ  γ  Gamma
Δ  δ  Delta
Ε  ε  Epsilon
Ζ  ζ  Zeta
Η  η  Eta
Θ  θ  Theta
Ι  ι  Iota
Κ  κ  Kappa
Λ  λ  Lambda
Μ  μ  Mu
Ν  ν  Nu
Ξ  ξ  Xi
Ο  ο  Omicron
Π  π  Pi
Ρ  ρ  Rho
Σ  σ  Sigma
Τ  τ  Tau
Υ  υ  Upsilon
Φ  φ  Phi
Χ  χ  Chi
Ψ  ψ  Psi
Ω  ω  Omega
OTHER SYMBOLS
→
Rightwards arrow: lim_{x→a}
↦
Rightwards arrow from bar: associate

Vertical bar: such that
:
Colon: such that
~
Tilde: asymptotically equal